The uniqueness of Leray-Hopf solutions is
considered as one of the most important issues in the mathematical fluid mechanics.
The well known Ladyžhenskaya-Prodi-Serrin condition states that,
the Leray-Hopf solution to NSE is regular and thus unique in the space \(L^\gamma _tL^p_x\) 
for \({2}/{\gamma }+{3}/{p}\le 1\)  with \(p>3\)  (see [1]}, [2]}, [3]}).
The endpoint case \(L^{\infty }_tL^3_x\)  was solved by Escauriaza-Seregin-Šverák [4]}.
The well-posedness of NSE for small initial data in the space \({\rm BMO^{-1}}\)  was proved by Koch-Tataru [5]}.
Moreover, Lions [6]} obtained the well-posedness for the hyperviscous NSE (REF ) when \({\alpha }\ge 5/4\) .
Concerning the generilized MHD system (REF ),
the uniqueness of global classical solutions
was proved by Wu [7]}
under the condition that \(\alpha _1, \alpha _2\ge {1}/{2}+{n}/{4}\) ,
which is exactly the Lions exponent for the hyperviscous NSE in dimension three \(n=3\) .
Moreover,
Zhou [8]} proved that,
when \({\alpha }_1={\alpha }_2\) ,
the solutions are smooth and unique in \(L^\gamma _tL^p_x\)  with \({2\alpha _1}/{\gamma }+{3}/{p}\le 2\alpha _1-1\) ,
which reduces to the Ladyžhenskaya-Prodi-Serrin condition for the NSE
when \(B\equiv 0\)  and \({\alpha }_1=1\) .
See also [9]}, [10]}, [11]}, [12]}, [13]} and references therein.
